Explorations in the theory of partition zeta functions

TL;DR

This paper introduces and surveys two types of partition zeta functions related to integer partitions, exploring their analytic properties, special formulas, and connections to modular forms and the Riemann Hypothesis.

Contribution

It provides new formulas, analytic continuations, and links to p-adic interpolation and the Riemann Hypothesis for these partition zeta functions.

Findings

Derived specialization formulas for partition zeta functions

Established analytic continuations and unusual formulas for the Riemann zeta function

Proved the Riemann Hypothesis for certain partition zeta polynomials

Abstract

We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the analytic continuations of these "partition zeta functions", find unusual formulas for the Riemann zeta function, prove identities for multiple zeta values, and see that some of the formulas allow for $p$-adic interpolation. The second family we study was anticipated by Manin and makes use of modular forms, functions which are intimately related to integer partitions by universal polynomial recurrence relations. We survey recent work on these zeta polynomials, including the proof of their Riemann Hypothesis.